5266
J . Phys. Chem. 1986, 90, 5266-521 1
Micelle Molecular Parameters from Surfactant Ionic Mobilities Robert L. Kay* and Kuo-Shin Leet Department of Chemistry, Carnegie-Mellon University, Pittsburgh, Pennsylvania I521 3 (Received: February 19, 1986; In Final Form: May I O , 1986)
Precise cation transference numbers for sodium and potassium dodecyl sulfate (DS) at 25 and 40 O C , respectively,are reported from moving boundary measurements using an improved design of radio frequency boundary detector. The concentration ranges studied were from dilute solutions to well into the micellar region. Precise conductance measurements are also reported for both salts. The critical micelle concentrations (cmc) obtained are 8.3 X and 8.2 X M for NaDS and 7.6 X and 7.5 X M for KDS from conductance and transference data, respectively. The results clearly conform to the Onsager limiting law for uni-univalent electrolytes. The limiting Walden product for the dodecyl sulfate ion shows no temperature dependence, indicating that the hydrophobic and hydrophilic properties of this long-chain anion tend to cancel When a phase separation model was adopted, it was possible to calculate a micellar aggregation number of 64, a micellar charge of -10.9, and an equivalent conductance of the micelle at 25 O C of 41 S cm2 equiv-’ from the concentration dependence of the transference number and the conductance data reported here and from known ion-selective-electrodeand micelle diffusion data.
1. Introduction Although precise conductances of surfactant solutions have been the subject of many investigations,’ precise transference data are scarce in spite of the fact that without such data anionic and cationic mobilities cannot be compared unambiguously and their temperature and pressure coefficients cannot be evaluated except at infinite dilution. This general lack of data is accounted for by the fact that the moving boundary method, the most precise method for measurements on dilute solutions, is generally not applicable to surfactant solutions because the boundaries are often not visible to the optical detectors in general use.2 Hartley et al.3 avoided this problem by using the indirect method2 in which the concentration of the following solution behind the boundary was determined by ac conductance measurements between small probe electrodes. They determined transference numbers for several cationic surfactants generally at concentrations above the critical micelle concentration (cmc). Owing to problems inherent to conductance measurements at extremely small electrodes and in dealing with impurities when making measurements on extremely small volumes of dilute solutions, this method is restricted to fairly concentrated solutions and to a precision of about 0.5%. Hoyer et al.4 reported a tracer electrophoretic method of determining ionic mobilities that was adapted by Mysels and DulinS to determine transference numbers for sodium and dodecyl sulfate (DS) ion constituents of NaDS. In this method, a trace amount of 22NaDS is placed in a horizontal tube containing a nonradioactive solution of NaDS. This tube is connected at its ends by suitable electrode compartments. After passage of a measured number of coulombs the radioactivity in the various compartments is determined, and the net amount lost by the horizontal tube per Faraday of current passed is a measure of the transference number of the sodium ion constituent. The method suffers from the lack of a restoring force2 such as that encountered in the moving boundary method and consequently is seriously affected by diffusion, electroosmosis, and temperature effects. Although every effort was made to reduce the effect of these disturbances, the method is capable of an accuracy of about 5%, although a precision of 0.3% is ~ l a i m e d . ~ Mukerjee6 reported transference numbers for NaDS at three concentrations below the cmc using the indirect method of Muir et ai.’ This method involves the removal of a portion of the NaDS solution that follows a NaCl solution in a typical moving boundary experiment. The concentration of this solution is automatically adjusted to a unique value so that the ’following” ion constituent moves with the same velocity as the ‘leading” ion constituent. This adjusted concentration was determined by conductance measurements on small sample volumes. Mukerjee, by developing Present address: Stauffer Chemical Co., Richmond. CA 94804.
0022-3654/86/2090-5266$01.50/0
meticulous techniques with extreme care, was able to attain a precision of 0.1 % in the transference numbers. Here, we report transference data for NaDS at 25 “ C from 3 to 25 X M. M and for KDS at 40 OC from 2 to 10 X Conductance data for both salts are also included. KDS was measured because of problems encountered with dilute solutions of NaDS which did not permit the Onsager limiting slopes to be verified unambiguously. 2. Experimental Section 2.1. Chemicals. Sodium dodecyl sulfate (NaCI2H,,SO4,British Drug House) was 99.0% pure. Among the three available suppliers (BDH, Research Plus, and Eastman Kodak), the BDH product gave the highest cmc in agreement with available and the best surface tension behavior. For example, a plot of surface tension (by the Du Noy ring method) as a function of concentration for the BDH product showed a small dip of only 2 dyn cm-’ at the cmc, and this is one-half to one-third of that obtained from the other two suppliers. Attempts to purify the BDH product further by recrystallization from a 50% H20-ethanol mixture did not affect the surface tension properties at the cmc nor did a Soxhlet extraction with petroleum ether for 48 h. Consequently, the NaDS (BDH) was used without further purification. Potassium dodecyl sulfate (KDS) was precipitated by cooling an aqueous solution of NaDS containing excess reagent-grade KCl followed by three recrystallizations from conductivity water. The Na content of the resulting product was found to be less than 0.2 ppm by atomic absorption spectroscopy. Distilled water was passed through a 5-ft mixed-bed ion-exchange column to produce water with a specific conductance of approximately 1 x lo-’ S cm-’. 2.2. Preparation of Solutions. Solutions, of NaDS were prepared by weight, and all weights were vacuum corrected. The (1) Mukerjee, P.; Mysels, K. J. Critical Micelle Concentration of Aqueous Surfactant Systems; Superintendent of Documents, U S . Government Printing
Office: Washington, DC, 1971; NSRDS-NBS, p 51. (2) Kay, R. L. In Electrochemistry; Yeager, E., Salkind, A. J., Eds.; Wiley: New York, 1973. (3) Hartley, G. S.; Collie, B.; Samis, C. S. Trans. Faraday SOC.1936, 32, 795. Samis, C. S.; Hartley, G. S. Ibid. 1938,’34, 1288. (4) Hoyer, H. W.; Mysels, K. J.; Stigter, D. J. J . Phys. Chem. 1954, 58, 385. ( 5 ) Mysels, K. J.; Dulin, C. I. J . Colloid Sci. 1955, 10, 461. (6) Mukerjee, P. J . Phys. Chem. 1958, 62, 1397. (7) Muir, D. R.; Graham, J. R.; Gordon, A. R. J . Am. Chem. SOC.1954, 76, 2157. (8) Lawrence, A. S. C.; Pearson, J. T. Trans. Faraday SOC.1967,63,495. (9) Tominaga, T.; Stem, T. B., Jr.; Evans, D. F. Bull. Chem. SOC.Jpn. 1980, 53, 795. (IO) Mukerjee, P.; Kapauan, P.; Meyer, H. G. J . Phys. Chem. 1966, 70, 783.
0 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 21, 1986 5267
Micelle Molecular Parameters
M.B. Cell
I I
I
Boloncing Network
I
I
R.F. Generotors
I
I
Lock-in
I I
Amplifier
I
I
Recorder
I I
I I
I I
I
I
I
Figure 1. Block diagram of the radio-frequency moving boundary detector.
molar concentrations were calculated from known density data for NaDS aqueous solutions.”*’2 The molecular weight used for NaDS was 288.38. Solutions of KDS were prepared similarly, except the solution densities were assumed to be the same as that of water at 40 OC (0.9922 g mL-3).’3 The molecular weight used for KDS was 304.49. 2.3. Apparatus. An “autogenic” moving boundary cell was used for all runs2 It consists of a Cd metal anode sealed onto one end of a 3-mm-I.D. tube which was connected to a Ag/AgCl cathode compartment at its other end. Initially the cell was filled with surfactant solution. On passage of an electrical current a boundary, formed between NaDS and Cd(DS)2solutions, moves up the tube with constant velocity. The operating principle of the redesigned radio-frequency moving boundary detector is essentially the same as that originally described by Pribadi’, but differs in several important features: (i) the operating frequency has been increased by a factor of 2 to 21.4 MHz; (ii) an entirely different type of balancing network has been developed; (iii) a commercial amplifier is used for phase-sensitive detection; and (iv) the input-output circuitry has been reversed. As shown in the block diagram in Figure 1 the circuit consists of five major parts: cell, balancing network, radio-frequency generators, lock-in amplifier, and recorder. The detection unit is connected to the transference cell through CT-C, and CR-C2, where the subscripts T and R designate transmitting and receiving, respectively. The moving boundary cell contained six probe sets each consisting of five metal bands about 1 mm apart. The metal bands originally14 were platinum films fused onto the outside walls of the tube, but these have been replaced with flattened wire twisted onto the tube and held in place with epoxy cement. The middle three,rings of each probe are part of a capacitance-resistance transformer bridge powered by a 21.4-MHz generator. The two outermost rings of each probe are grounded and act as guard rings, isolating each probe set from its neighboring probes. Without these guard rings extremely unsymmetrical boundary traces are obtained. The bridge is powered through the center ring, and the (1 1) Franks, F.; Quickenden, M. J.; Ravenhill, R. J.; Smith, H. T. J. Phys. Chem. 1968, 72, 2668. (12) Musbally, G . M.; Perron, G . ; Desnoyers, J. E. J . Colloid Interface Sci. 1974, 48, 494. (13) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Butterworths: London, 1959; p 457. (141 Pribadi. K . S . J . Solution Chem. 1972, 1,455. Pribadi, K. S. Ph.D. Thesis; Carnegie-Mellon University, 197 1.
resultant output from the adjacent rings is sent to the detector. This is the reverse of the original design, but this simple change has increased the signal-to-noise ratio by a factor of 2 and has substantially reduced the range required by the balancing network. This is explained by the fact that if the transformer bridge is powered through the second and fourth rings, half the current is sent to the adjacent grounded guard rings. A new feature substantially different from the earlier circuit is the balancing network. Rather than using a variable capacitor-resistor series network that is restrictive in range, two double-balanced mixersI5 (M3 and M4 in Figure 1) were employed as current-controlling attenuators. In this way, the amount of in-phase and out-of-phase signal can be controlled by simply adjusting the input dc current to the mixers at their I F port. Mixers have much better temperature and long-term stability than a capacitor-resistor network and permit long-range operation. The operating principle of the circuit is basically simple. The generator G I transmits a 21.4-MHz signal to the center ring of each probe. At the start of a run, with a uniform solution in each probe region, the error signal from the second and fourth ring of each probe is transmitted to the detector through transformer T I , and its magnitude is set to any convenient level (the zero balance) by the mixers M3 and M4 acting through transformer T,. This error signal is amplified by a tuned rf receiver-amplifier A1 and the resulting signal mixed in MA1 with a 21.405-MHz signal from generator G2. The sum frequency is removed by filtering, and the difference (audio) frequency “signal” of 5 kHz is fed into the lock-in amplifier. A “reference” with exactly the same frequency as the “signal” is obtained by mixing, at MA2, the rf signals from generators G1 and G2. In the lock-in amplifiers the signal and the reference are amplified to a maximum gain of 20 000 and fed into a phase-sensitive detector. The phase of the reference is adjusted at P2 to give a recordable dc signal after amplification at 53. Either a commercial or simple homemade lock-in amplifier can be used. Further details of the circuit have been reported by Lee.’6 In actual operation the detector is first balanced to the lowest possible output at J1 with uniform solution between all probes, by adjusting I3 and I,. For optimum results it is necessary to adjust the phase shifter P2 in the lock-in amplifier. It has been found from experience that, for most solutions, in-phase operation gives the best results, although for solutions of extremely high or low resistance 90° out-of-phase operation was found preferable. Once (15) Mini-circuits Laboratory, Catalog 6, p 21, 1974. (16) Lee, K. Ph.D. Thesis, Carnegie-Mellon University, 1981
5268
The Journal of Physical Chemistry, Vol. 90, No. 21, 1986
TABLE I: Molar Conductances of NaDS and KDS in Aqueous Solution NaDS, 25 OC KDS, 40 'C 104c 23.30 40.53 60.90 76.83 94.68 111.55 131.58 148.02 170.48
A 69.34 68.36 67.47 66.8 1 61.84 55.50 51.45 48.42 45.24
Kay and Lee TABLE 11: Observed and Corrected Transference Numbers for NaDS at 25 OC corrections
104c
A
104c
7.Cnhr?
solvent
vol
r
9.7851 29.695 49.574 64.399 78.798 94.391 114.28
123.14 120.51 118.94 118.07 114.50 102.72 91.14
30 35 40 50 60 70 75 80 85 90 95 100 125 150 175 200 225 250
0.6805 (8) 0.6876 (4) 0.6903 (1) 0.6942 (1) 0.6971 (2) 0.6971 (2) 0.6970 (7) 0.6943 (3) 0.6792 (3) 0.6548 (2) 0.6239 (2) 0.5956 (1) 0.4t 50 (7) 0.3530b 0.2650 (1) 0.1873 (3) 0.1226 ( I ) 0.0688 (2)
0.0003 0.0003 0.0003 0.0002 0.0002 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001
-0.0002 -0.0002 -0.0002 -0.0003 -0.0003 -0.0004 -0.0004 -0.0005 -0.0005 -0.0006 -0.0007 -0.0008 -0.0015 -0.0022 -0.0029 -0.0037 -0.0045 -0.0054
0.6806 0.6877 0.6904 0.6941 0.6970 0.6969 0.6967 0.6939 0.6788 0.6543 0.6233 0.5949 0.4646 0.3508 0.2621 0.1836 0.1181 0.0634
the phase is optimized and the circuit rebalanced, voltage peaks of nearly a volt can be recorded during the passage of an NaDS boundary through a probe. The ac conductance apparatus and techniques are essentially the same as described el~ewhere.".'~ The Dyke-Jones bridge (Leeds and Northrup) was used for the resistance measurements. The electrodes of the Kraus-type Erlenmeyer cell were lightly platinized. The cell was cleaned with steaming nitric acid. The cell constant was determined at 25 OC ( J = 1.3936 cm-') by measuring the conductance of KCl in water and using the averaged conductance parameters suggested by Fuoss et al.I9 The cell constant calculated for 40 "C was 1.3935 cm-'.I9 An all-glass, closed apparatus was used to introduce solvent into the cell without contamination from the atmosphere, and a Hawes-Kay salt cup dispensing device18+z0 was used to introduce the surfactant into the solvent in the closed cell. The light-weight transformer oil (Gulf Transcrest 62) in the constant-temperature bath was regulated to fO.OO1 OC at both 25 and 40 OC by means of a mercury-on-glass thermoregulator and an automatic proportional heater (Tronac). The absolute temperature was set with a platinum resistance thermometer.
3. Results The molar conductances A (S cm2 mol-') and molar concentrations C (mol dm-3) are given in Table I for NaDS at 25 OC and KDS at 40 "C. For the purpose at hand it was sufficient to extrapolate to infinite dilution with the Onsager limiting slope.2' Only one run was attempted in each case, and the measurements were not extended to extremely dilute solutions as is necessary for an accurate determination of A,. Our values of the limiting molar conductances A, were 72.50 and 126.62 for NaDS and KDS, respectively. The value of NaDS can be compared to 72.93 obtained by Parfitt and SmithZZextrapolated by using a FuossOnsager extended conductance equation. Since this treatment always gives a somewhat higher value than the limiting Onsager equation, we consider our results satisfactory. We are unaware of any other determination of A,(KDS). The cmc of 8.2, X IOw3 M obtained for NaDS is in good agreement with accepted values,' particularly with the 8.2, X M value reported by Mukerjee et a1.I0 The cmc of 7.6 X lo3 M for KDS at 40 "C agrees favorably with a previous determination of 7.8 X IO3 M.z3 The transference numbers P for the sodium ion constituent2 were calculated from
#The numbers in parentheses are the standard deviations (in the last digit quoted) of multiple runs at different currents. bSingle determination. TABLE 111: Observed and Corrected Cationic Transference Numbers for KDS at 40 O C corrections
104c 20 30 40 50
60 81 90 100
r,,,
solvent
vol
r
0.7616 (1) 0.7636 (3) 0.7654 (1) 0.7652 (4) 0.7643 (3) 0.7392 (2) 0.6875 (6) 0.6360 (2)
0.0030 0.0023 0.0014 0.0012 0.0009 0.0008 0.0007 0.0006
-0.0002 -0.0002 -0.0003 -0.0003 -0.0004 -0.0006 -0.0007 -0.0009
0.7644 0.7657 0.7665 0.766 1 0.7648 0.7694 0.6875 0.6357
Here, C is the molar concentration of the leading solution, NaDS and KDS in our case, V the volume swept out by the boundary by the passage of a current i (A) for a time t (s), and F is the Faraday constant. The term in the square bracket is the solvent conductance correction where K~ and K , are the solvent and solute specific conductivities, respectively. This correction limits the precision of the measurement for dilute solutions. The last term in eq 1 is the volume correction which limits the precision of the measurement for concentrated solutions. AV is the volume change between the boundary and the closed anode compartment, resulting from the passage of electrical current. For the NaDS system AV is given by eq 3 where the molar volume A V = j/,vCd(DS),] - i/zV(Cd) - P(NaDS)V(NaDS)
V(Cd) = 13.0 cm3z4 and the partial molar volume V(NaDS) = 238 cm3 for concentrations less than 0.008 MI' and 255 cm3 for more concentrated solutions.l2 The partial molar volume of Cd(DS)2 was estimated from eq 4. Comparable equations for vCd(DS),] = 2V(NaDS)
where p & d
= 10-3CVF/it
(2)
(17) Kay, R. L.; Zawoyski, C.; Evans, D. F. J . Phys. Chem. 1965, 69, 3878, 4208.
(18) Kay, R . L.; Hales, B. J.; Cunningham, G. P. J . Phys. Chem. 1967, 71, 3925.
(19) Lind, J. E., Jr.; Zwolenck, J. J.; Fuoss, R. M. J . Am. Chem. SOC. 1959.81. 1557. (20) Hawes, J. L.; Kay, R. L. J . Phys. Chem. 1965, 69, 2420. (21) Reference 13, p 468. (22) Parfitt, G.D.; Smith, A. L. J . Phys. Chem. 1962, 66, 942. ( 2 3 ) Meguro, K.; Kondo, T.; Yoda, 0. J . Chem. SOC.Jpn., Pure Chem. Sec. 1956, 7 7 , 1236.
(3)
+ V(Cd2+) - 2V(Na+)
(4)
KDS can be written and the partial molar volumes estimated from known volume data for potassium saltszs and by assuming comparable volumes of micellization for NaDS and KDS. The experimental data and corrected transference numbers are given in Tables I1 and I11 at rounded values of the concentrations obtained from larger plots of the experimental data. Each value is the average of two or more runs at different currents. Variations of up to 100%in the electrical current for the dilute solution runs and 15% for the more concentrated solutions did not affect the results significantly. The upper concentration limit of 0.025 M for NaDS was determined by the extremely low boundary ve(24) MacInnes, D. A,; Longsworth, L. G. Chem. Rea. 1932, 1 1 , 171.
( 2 5 ) Millero, F. J. Chem. Reo. 1971, 71, 147.
The Journal of Physical Chemistry, Vol. 90,No. 21, 1986 5269
Micelle Molecular Parameters
\-I
_____--L.L.
t
T+
O.*
0
a
o This work
m Mysels and Dulin
t
a
.65
a,&
I
,
I
0.04
I
I
I
I
0.12
0.08
I
T + .70
>\,I
0.16 \\
I
1
0.20
KDS - H20-40°C
.02
Figure 2. Transferencenumbers for the sodium ion constituent in NaDS solutions at 25 "C.
.68-
T+
.66-
.64-
I
I0
oThis work Mukerjee
A
NO DS-HzO - 25'C
( C,mole 1-1)
To+(NaDS) =
Xo(Na+) Ao(NaDS)
(5)
Xo(Na+) = 50.2026 and Ao(NaDS) = 72.9322are known. The resulting To+(NaDS) = 0.6883 is plotted as the open square in Figure 2 and the Onsager limiting slope (LL)*' drawn through the point. It can be seen that our values of T+ approach the limiting law at the cmc and agree with the general trend of the data of Mysels and DulinS at higher concentrations, although the precison is now much improved. Our data appear to extrapolate to their data point beyond C = 0.029 where T becomes negative. A problem was encountered with dilute solutions of NaDS. As the concentration decreases from the cmc, the transference numbers decrease significantly from the Onsager limiting slope as can be seen clearly in Figure 3. Also, the measured T+values show significant current dependence at the lowest concentration measured. This result can be attributed possibly to density instability across the boundary, but we consider it more likely that (26) Gunning, H. E.; Gordon, A. R. J . Chem. Phys. 1942, 10, 126. (27) Kay, R. L.; Evans, D. F. J . Phys. Chem. 1966, 70, 2325.
b
(C,mole 1.1) I f z I
I
I
.04
I
I
.06
i
l
I
.08
I
.IO
Figure 4. Transference numbers for the potassium ion constituent in KDS at 40 "C.
the mobility of the leading sodium ion constituent is not sufficiently greater than that of the leading cadmium ion constituent, a necessary condition for the formation of stable boundaries. In aqueous solutions, for example, CdClz forms a good following solution for NaCl only because of the formation of the C d C P complex ion which lowers the mobility of the Cd ion constituent. If the cadmium solution is below its cmc in the concentration range in question, the effect could be attributed to the inability of the DS- ion to enter the coordination sphere of the cadmium ion to form a complex ion. However, the cmc for many divalent dodecyl sulfates,' and Zn(DS)z and Pb(DS)* in particular,28must be close M, which is well below the adjusted "Kohlrausch" to 1 X concentration of our following solution. Consequently, it would appear that micelles are present in the following solutions of all our experiments. A detailed analysis of this effect is not possible until more is known about the micellar properties of Cd(DS)2 solutions. It is interesting to note the results obtained by Mukerjee for NaDS as mentioned in the Introduction. His three points (shown by the triangles in Figure 3) are in excellent agreement with the Onsager limiting law as we have drawn it and confirm our contention that our data do approach the limiting law at about 0.006 M. Since the data clearly conform to the limiting law for a 1 :1 electrolyte, it appears that the assumption of dimer formation6 is not necessary to account for the concentration dependence of the transference numbers. In order to test this hypothesis further, we extended our measurements to KDS. The mobility of the Kf ion is much greater than that of the cadmium ion constituent, whereas the density difference should be even less than that obtained with the sodium salt. As can be seen in Figure 4, the cation transference numbers clearly approach the limiting law given by the straight line, and there is no sign of negative departures from this line down to 0.002 M. This confirms our contention that it is a relative mobility problem that affects our NaDS results and not a density problem. The measurements of KDS were carried out at 40 OC because the solubility of KDS at 25 OC was below it cmc. To establish the position of the Onsager limiting slope in Figure 4,it was necessary to calculate To+= 0.7568 from an analogue of eq 5 for KDS using Xo(Kf, 40 "C) = 95.82 obtained by interpolation of known data26and from hoobtained in this work. We can also calculate Xo(DS-) of 22.7 and 30.8 and Walden products of 0.202 and 0.201 at 25 and 40 "C, respectively. Thus, the Walden product of the DS- ion has a near zero temperature coefficient, and consequently, the hydrophilic and hydrophobic properties of this long-chain anion appear to cancel one a n ~ t h e r . ~ ' It is possible to analyze the transference data at concentrations above the cmc to obtain useful properties of the micelle such as its conductance, charge, and aggregation number. The concentration dependence of the transference number of the sodium ion (28) Miyamoto, S. Bull. Chem. Soe. Jpn. 1960, 33, 371.
5270 The Journal of Physical Chemistry, Vol. 90, No. 21, 1986
Kay and Lee
TABLE I V Fraction of Free Na' and DS-Ions from Equations 11 and 12 and the Equivalent Conductance of the Micelle
0.0125 0.0150 0.0175 0.0200 0.0225 0.0250
24.67 16.94 11.74 7.78 4.72 2.44
0.703 0.620
0.560 0.515 0.481 0.454
25.2 29.4 30.7 31.5 32.1 32.3
28.3 31.3 33.2 34.6 35.3 36.3
0.513 0.416 0.347 0.295 0.255 0.223
38.0 40.5 41.1 41.5 41.3 41.6
constituent in terms of the species i present is given by the Spiro equation,29which for this system becomes eq 6 where C and X (cA)Na+ - m c M ( A M / z M ) TNa = (6) CICLXI
->
E")
w
135
I
I
I
I
I
1
I
I
I
1
2
3
4
5
6
7
8
9
I
I
I
I
I
5
6
7
8
9
I!
IO2 c?
Figure 5. Plot of eq 16 for the Na' ion.
are the molar ionic concentrations and conductances, respectively, Z is the absolute magnitude of the ionic charge, the subscript M indicates the micelle, and m is the number of sodium ions bound per micelle. A material balance for the Naf ion in terms of C,, the stoichiometric concentration of NaDS, gives eq 7 which rearranges to eq 8 where a is the fraction of free Na+ ions; Le., (Y c[= c N a + + mCM (7) CM = (ct/m)(l - cy) (8) = CN,+/C,. Substitution of this expression for CMinto eq 6 and rearrangement produces eq 9 where AM' = AM/ZMis the equivalent conductance of the micelle. Likewise, a material balance for the TNa+ANaDS = a X N a + - ( I - a)hM' (9) DS- ion gives CM = (C,/n)(1 - 4) where n is the number of DSions per micelle (the aggregation number) and 4 is the fraction of free DS- ions. Substitution in eq 6 produces eq IO. TDS-ANaDS = dXDS- + (1 - +)AM' (10) We obtained cy and 6 from emf data measured with Na+ ion and DS- ion selective electrodes. For this purpose we used the actual experimental data available in supplemental material asA second set of reliable sociated with a paper by Kale et measurements for these systems have been reported by Cutler et al.,31but the actual experimental data were not included in the publication. Kale et al. make the claim that the two sets of data are similar. Kale et al. fitted their measured emfs at NaDS concentrations below the cmc to a Nernst type equation of the form E* = Eo A N log Cty,
where E, and N were treated as parameters and the activity coefficients were assumed to be approximated by log y+ = -0.509C1/'/( 1 + C1/'). Using their emf values for the concentration range 4 X loe4 I C I 5.88 X (concentrations well below the cmc) and applying least squares, we obtained for the Na+ ion eq 11 and for the DS- ion eq 12. Although the DS- ion Ef(mV) = Eof + N+ log C,y+ = 126.3 f 2.0 (53.8 h 0.7) log CNa+y+(11)
+
E-(mV) = Eo- - N- log Cty* = -96.9 A 1.7 - (59.7 f 0.6) log CDs-y* (12) electrode gives a Nernstian response within experimental error, N f for the Na' ion is well below the Nernst value of 59 mV, a matter of some concern. This aspect of the problem is considered in more detail in what follows. CN,+ and CDs-(and consequently a and 4 ) were evaluated from eq 11 and 12 for the micellar region by assuming the mean ion activity coefficient for the concentration range 0.01 IC, I0.025 ~
~~
(29) Spiro, M. In Physical Chemistry of Organic Soluent Systems; Covington, A. K., Dickinson, T., Eds.; Plenum: London, 1973; Chapter 5, p 617 (30) Kale, K. M.; Cussler, E. L.; Evans, D. F. J . Phys. Chem. 1980, 84, 593. (31) Cutler, S. G.; Meares, P.; Hall, D. G. J . Chem. Soc., Faraday Trans. 1 1978, 74, 1758. ( 3 2 ) Degiorgio, V.; Corti, M. J . Colloid Interface Sci. 1984, 101, 289.
-95
I
1
I
I
1
2
3
4
IOPcl" Figure 6. Plot of eq 17 for the DS- ion.
is given by eq 13 with the ionic strength estimated from eq 14 using ZM= 10.9 and CMevaluated from eq 8 with m = 53.1,
+
log y* = -0.5091'/2/(1 1'/2)
(13)
I = O.5((*.Ct+ +Ct + CMZM')
(14) acceptable values for these two quantities as will be shown in what follows. Two independent values of AM' can be evaluated, one from a and eq 9 and one from 4 and eq 10 as given in the fourth and seventh columns, respectively, of Table IV. For this purpose we estimated ANa+ and ADS- by assuming the limiting Onsager equation was valid so that XNa+ = 50.2 - 41.81'f2 and ADS- = 22.7 - 35.21'1'. Two aspects of the two sets of AM' values in Table IV require consideration. First, there is a sharp decrease of AM' at lower surfactant concentrations which we can show is the result of experimental error (see Figure 7). Consequently, we assume AM' to be independent of the NaDS concentration over the narrow range studied here. Second, and more importantly, it appears that AM' calculated from the sodium ion electrode data is almost 25% lower than AM' calculated from the DS- ion electrode data, and it will be shown in what follows that the AM' value obtained from the DS- ion data leads to acceptable values of the charge and aggregation number for the micelle. Consequently, the non-Nernstian response of the sodium ion electrode becomes suspect as the source of the problem. This focuses attention on eq 11 and 12 in which Eo and N are treated as adjustable parameters and an arbitrary assumption is made concerning the magnitude of the activity coefficients. We prefer a more traditional approach by defining the quantities E'' and E-' such that E+'(mV) E-'(mV)
E+ - 59.0 log Cv,+ E-
+ 59.0 log CDS-
(15) In Figure 5 Et' is plotted vs. (Ct)'/'. The sodium ion gives a straight line from about M to the cmc such that E
E" = 143.2 - 94.4(Ct)'/2
(16) but at lower concentrations the deviations from linearity are large. The limiting Debye-Huckel equation for the activity coefficient would require a slope of about 29.5 in place of the 94.4 found, indicating clearly the magnitude of the non-Nernstian response
The Journal of Physical Chemistry, Vol. 90, No. 21, 1986 5271
Micelle Molecular Parameters TABLE V Fraction of Free Na+ and DS- Ions from Equations 18 and 19 and the Equivalent Conductance of the Micelle from CN,t from CDs0.0125 0.01 50 0.0175 0.0200 0.0225 0.0250
0.738 0.665 0.615 0.578 0.551 0.530
0 01
34.5 39.4 41.4 42.6 43.8 44.2
0.508 0.41 1 0.342 0.291 0.251 0.219
38.0 40.3 41.0 41.4 41.2 41.5
0020
0015
0025
ct
Figure 7. Dependence of the micelle equivalent conductance AM‘ on the NaDS concentration: (0)from eq 11, ( 0 )from eq 16, (0)from eq 12 and 17.
obtained with this electrode. The same procedure applied to the DS- ion electrode produces the results shown in Figure 6 and in eq 17. The DS- ion electrode produces a plot with a slope much
E-’ = -94.3
+ 16.0(Ct)’/2
(17) closer to that expected from the limiting law and eq 17 does fit the data over the complete concentration range from low4M to the cmc. It is also clear that this electrode detects the approaching cmc at a lower concentration of surfactant than the Na+ ion electrode. If eq 16 and 17 hold in the micellar region, CNat and CDs-(and, consequently, a and 4) for that region can be calculated from eq 18 and 19 without the need for any assumption concerning the
+ 94.4Z1/’)/59.0 - 94.3 + 16.0Z1/2)/59.0
log
cNat
= (E+ - 143.2
(18)
log
CDS-
= (-E-
(19)
form of the activity coefficient function. Substitution of these values into eq 9 and 10 along with the previously calculated values of ANa+ and ADS- produces the two sets of AM’ given in Table V. The agreement in AM’ resulting from the data from the two different electrodes and the transference numbers now are considerably improved and almost within the experimental error as shown by the error bars in the plot of AM’ in Figure 7. However, AM’ from the Na’ electrode data still show a large concentration dependence, whereas AM’ from the DS- electrode data appear to reach a constant value above C, = 0.02 M and give the same values of AM’ from eq 12 and 17, namely 41.1 f 0.5. If we assume that Stokes’ law holds for the micelles at the ionic strengths involved, the charge on the micelle can be estimated from AM’ and the diffusion coefficient DM for the micelles since
and, consequently, ZM
= 0.266 X lO*AM’/DM
(21)
Establishing a reliable value of DM is complicated. Although the data from dynamic light ~ c a t t e r i n g ,Taylor ~~ di~persion,~~ (33) Rhode, A,; Sackmann, E. J . Phys. Chem.1980, 84, 1598. (34) Weinheimer, R. M.; Evans, D. F.; Cussler, E. L. J . Colloid Interface Sci. 1981, 80,357.
c o n d u c t i ~ i t y and , ~ ~ boundary spreading36 measurements are in reasonable agreement, these methods produce DM that increase unrealistically with increasing NaDS concentration for the concentration range under consideration here. Further, DM assumes more reasonable behavior only in the presence of a large excess of salt. On the other hand, the older data of Stigter et aL3’ from diffusion measurements on micelles tagged with a solubilized dye showed a reasonable small decrease in DM with increasing NaDS concentration above the cmc. This result was verified by Weinheimer et al.34by Taylor diffusion on surfactant solutions tagged with a solubilized dye. It would appear, as pointed out by Weinheimer et al., that the different techniques measure an average over the various species present. We are of the opinion that the methods involving dye give the most reliable results since in such measurements the movement of the dye is followed and the dye is contained only in the micelles. It is true that the presence of the dye can affect the properties of the micelle, but this effect can be small as has been shown from recent studies of the inclusion of pyrene in micelles.38 In any case, since the diffusion data from all the methods appear to extrapolate to about 1.O X lo6 cm2 s-l at the cmc, we have accepted that value for DM for NaDS in the concentration range studied here. Substitution of this value in eq 21 predicts Z = 10.9 for the micellar charge. In order to obtain the aggregation number of the micelles from eq 21 from this value of Z , it is necessary to obtain another relationship between n and m besides the known relation Z = n - m. This can be accomplished by assuming a phase separation model for micelle formation in which the micelle is assumed to be a separate phase of constant activity. On this basis, the condition for equilibrium becomes eq 22 where a is the activity of n log aDB m log aNat = constant (22) the species indicated. A plot of log aDs-vs. log aNat should be linear with slope -R where R = m/n. Weinheimer et al.34have summarized the results of a variety of methods that made use of eq 22 producing an average R = 0.83 with an standard deviation of 0.03. n could then be calculated from eq 23 which gives n = 64 f 11. n = Z/(1 - R) (23) K r a t ~ h v i has l ~ ~reviewed the many determinations of n for this system and concluded that the results obtained from the light scattering measurements of Huisman40 are the most reliable. Huisman reports n = 58 for aqueous NaDS solutions. More recently Lianos and Zana38report n = 64 from fluorescence decay studies on NaDS micelles containing solubilized pyrene. Their measurements indicated a constant value of n from the cmc to 0.3 M, and their data for solutions containing NaCl are in good agreement with the Huisman data for similar solutions. Our value of n is in excellent agreement with these values although our result has an uncertainty of about 20% resulting primarily from the term 1 - R in eq 23 which magnifies the error in m / n by a factor of 5. It should be possible to reduce this error substantially by determining the ratio m / n from extremely precise conductance measurements in the vicinity of the cmc. Thus, it appears that transport properties can provide structural data for micellar solutions that are consistent with those obtained from other methods. The use of transference numbers permits molecular information to be obtained from counterion selective electrode data, thereby eliminating the need for the electrodes reversible to the surfactant ion which are often not readily available.
+
Acknowledgment. This research was supported by Grant C H E 7713 137 from the National Science Foundation. We acknowledge the considerable assistance of K. Pribadi with the redesign of the radio-frequency moving boundary detector. (35) Leaist, D. G . J . Colloid Interface Sci., in press. (36) Kratohvil, J. P.; Aminabhavi, T. M. J . Phys. Chem. 1982.86, 1254. (37) Stigter, D.; Williams, R. J.; Mysels, K. J. J. Phys. Chem. 1955, 59, 330. (38) Lianos, P.; Zana, R. J . Colloid Interface Sci. 1981, 84,100. (39) Kratohvil, J. J . Colloid Interface Sci. 1980, 75,271. (40) Huisman, H. F. Proc. K.Ned. Akad. Wet., Ser. B: Phys. Sci. 867, 1964, 367, 376, 388, 407.
